Let \(\lbrace B(x_i,r)\rbrace _{i=1}^N\)  be a cover of \(E\cap B(x,2R)\) , with
\(N=N\bigl (E\cap B(x,2R),r\bigr )\) . Then
\(E_r\cap B(x,R)\subset \bigcup _{i=1}^N B(x_i,2r),\) 
and thus
\(\vert E_r\cap B(x,R)\vert \le C(n) N(2r)^n= C_2(n)r^n N\bigl (E\cap B(x,2R),r\bigr ).\) 
This proves the second inequality in the claim.
Conversely, let
\(\lbrace B(x_i,r)\rbrace _{i=1}^N\)  be a cover of \(E\cap B(x,R/2)\) 
such that \(x_i\in E\cap B(x,R/2)\)  for all \(i=1,\ldots ,N\)  and the balls
\(B(x_i,r/2)\)  are pairwise disjoint (such a cover can be found by
choosing \(\lbrace x_i\rbrace _{i=1}^N\)  to be a maximal \(r\) -net in
\(E\cap B(x,R/2)\) , see [1]}).
Then
\(E_r\cap B(x,R)\supset \bigcup _{i=1}^N B(x_i,r/2),\) 
and thus
\(\vert E_r\cap B(x,R)\vert \ge C(n) N(r/2)^n\ge C_1(n) r^n N\bigl (E\cap B(x,R/2),r\bigr ).\) 
This proves the first inequality in the claim.
